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minimalPresentation(Ring) -- compute a minimal presentation of a quotient ring

Synopsis

Description

The computation is accomplished by considering the relations of R. If a variable occurs as a term of a relation of R and in no other terms of the same polynomial, then the variable is replaced by the remaining terms and removed from the ring. A minimal generating set for the resulting defining ideal is then computed and the new quotient ring is returned. If R is not homogeneous, then an attempt is made to improve the presentation.
i1 : R = ZZ/101[x,y,z,u,w]/ideal(x-x^2-y,z+x*y,w^2-u^2);
i2 : minimalPresentation(R)

      ZZ
     --- [x, u, w, MonomialOrder => GRevLex => 3]
     101
o2 = --------------------------------------------
                         2    2
                        u  - w

o2 : QuotientRing
i3 : R.minimalPresentationMap

          ZZ
         --- [x, u, w, MonomialOrder => GRevLex => 3]
         101                                                   2       3  
o3 = map(--------------------------------------------,R,{x, - x  + x, x  -
                             2    2
                            u  - w
     ------------------------------------------------------------------------
      2
     x , u, w})

              ZZ
             --- [x, u, w, MonomialOrder => GRevLex => 3]
             101
o3 : RingMap -------------------------------------------- <--- R
                                 2    2
                                u  - w
i4 : R.minimalPresentationMapInv

            ZZ
           --- [x, u, w, MonomialOrder => GRevLex => 3]
           101
o4 = map(R,--------------------------------------------,{x, u, w})
                               2    2
                              u  - w

                     ZZ
                    --- [x, u, w, MonomialOrder => GRevLex => 3]
                    101
o4 : RingMap R <--- --------------------------------------------
                                        2    2
                                       u  - w

See also